TREE AUTOMATA BASED ON COMPLETE RESIDUATED LATTICE-VALUED LOGIC: REDUCTION ALGORITHM AND DECISION PROBLEMS

In this paper, at first we define the concepts of response function and accessible states of a complete residuated lattice-valued (for simplicity we write $\mathcal{L}$-valued) tree automaton with a threshold $c.$ Then, related to these concepts, we prove some lemmas and theorems that are applied in considering some decision problems such as finiteness-value and emptiness-value of recognizable tree languages. Moreover, we propose a reduction algorithm for $\mathcal{L}$-valued tree automata with a threshold $c.$ The goal of reducing an $ \mathcal{L}$-valued tree automaton is to obtain an $\mathcal{L}$-valued tree automaton with reduced number of states %that all of its states are accessible all of which are accessible, in addition it recognizes the same language as the first one given. We compare our algorithm with some other algorithms in the literature. Finally, utilizing the obtained results, we consider some fundamental decision problems for $\mathcal{L}$-valued tree automata including the membership-value, the emptiness-value, the finiteness-value, the intersection-value and the equivalence-value problems.